Singleton-preserving predicate liftings should not be confused with
Kurz and Leal's singleton
liftings [1]}, [2]}. The
definition of the latter is based on the one-to-one correspondence
between subsets of \(T( 2^n )\)  and \(n\) -ary predicate liftings for
\(T\)  [3]}, which maps an \(n\) -ary predicate lifting
\(\lambda  \)  to
\(\lambda  _{2^n}( \pi _1^{-1}(\lbrace \top \rbrace ),\ldots ,\pi _n^{-1}(\lbrace \top \rbrace ) )\subseteq T( 2^n )\) ,
and \(C\subseteq T(2^n)\)  to the lifting \(\lambda  \)  defined by
\(\lambda  _{X}( A_1,\ldots ,A_n ) = \lbrace  t\in TX\mid T\langle \chi _{A_1},\ldots ,\chi _{A_n}\rangle (t) \in C\rbrace \) ,
where \(\pi _i: 2^n \rightarrow 2\)  is the \(i\) -th projection and
\(\chi _A: X \rightarrow 2\)  denotes the characteristic function of
\(A\subseteq X\) . An \(n\) -ary predicate lifting is a singleton
lifting if it corresponds to a singleton subset of \(T(2^n )\) .
